ON DETERMINING THE DISTANCE SPECTRUM OF A CLASS OF DISTANCE INTEGRAL GRAPHS

The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix‎‎D(G)‎. ‎A graph is called distance integral if all of its‎‎distance eigenvalues are integers.‎‎Let n and k be integers with n>۲k‎, ‎k\geq۱‎. ‎The bipartite Kneser graph H(n,k) is the graph with the set of all k and n-k subsets of the set [n]=\{۱,۲,...,n\} as vertices‎, ‎in which two vertices are adjacent if and only if one of them is a subset of the other‎. ‎In this paper‎, ‎we determine the distance spectrum of H(n,۱)‎. ‎Although the obtained result is not new \cite{۱۲}‎, ‎but our proof is new‎. ‎The main tool that we use in our work is the orbit partition method in algebraic graph theory for finding the eigenvalues of graphs‎. ‎We introduce a new method for‎‎determining the distance spectrum of H(n,۱) and show how‎‎a quotient matrix can contain all distance eigenvalues of‎‎a graph.‎

کلیدواژه ها:

Distance matrix ، distance spectrum ، orbit partition ، bipartite Kneser graph